Theorem pnt 22862

Description: The Prime Number Theorem: the number of prime numbers less than x tends asymptotically to x / log ( x ) as x goes to infinity. This is Metamath 100 proof #5. (Contributed by Mario Carneiro, 1-Jun-2016.)

Assertion

Ref	Expression
pnt	|- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1

Proof of Theorem pnt

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1re 9384	. . . . . . 7 |- 1 e. RR
2	1	rexri 9435	. . . . . 6 |- 1 e. RR*
3		1lt2 10487	. . . . . 6 |- 1 < 2
4		df-ioo 11303	. . . . . . 7 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
5		df-ico 11305	. . . . . . 7 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
6		xrltletr 11130	. . . . . . 7 |- ( ( 1 e. RR* /\ 2 e. RR* /\ w e. RR* ) -> ( ( 1 < 2 /\ 2 <_ w ) -> 1 < w ) )
7	4, 5, 6	ixxss1 11317	. . . . . 6 |- ( ( 1 e. RR* /\ 1 < 2 ) -> ( 2 [,) +oo ) C_ ( 1 (,) +oo ) )
8	2, 3, 7	mp2an 672	. . . . 5 |- ( 2 [,) +oo ) C_ ( 1 (,) +oo )
9		resmpt 5155	. . . . 5 |- ( ( 2 [,) +oo ) C_ ( 1 (,) +oo ) -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) )
10	8, 9	mp1i 12	. . . 4 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) )
11	8	sseli 3351	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> x e. ( 1 (,) +oo ) )
12		ioossre 11356	. . . . . . . . . . 11 |- ( 1 (,) +oo ) C_ RR
13	12	sseli 3351	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> x e. RR )
14	11, 13	syl 16	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> x e. RR )
15		2re 10390	. . . . . . . . . . 11 |- 2 e. RR
16		pnfxr 11091	. . . . . . . . . . 11 |- +oo e. RR*
17		elico2 11358	. . . . . . . . . . 11 |- ( ( 2 e. RR /\ +oo e. RR* ) -> ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x /\ x < +oo ) ) )
18	15, 16, 17	mp2an 672	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x /\ x < +oo ) )
19	18	simp2bi 1004	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 2 <_ x )
20		chtrpcl 22512	. . . . . . . . 9 |- ( ( x e. RR /\ 2 <_ x ) -> ( theta ` x ) e. RR+ )
21	14, 19, 20	syl2anc 661	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) e. RR+ )
22		0red 9386	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 0 e. RR )
23	1	a1i 11	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 1 e. RR )
24		0lt1 9861	. . . . . . . . . . . 12 |- 0 < 1
25	24	a1i 11	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 0 < 1 )
26		eliooord 11354	. . . . . . . . . . . 12 |- ( x e. ( 1 (,) +oo ) -> ( 1 < x /\ x < +oo ) )
27	26	simpld 459	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 1 < x )
28	22, 23, 13, 25, 27	lttrd 9531	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> 0 < x )
29	13, 28	elrpd 11024	. . . . . . . . 9 |- ( x e. ( 1 (,) +oo ) -> x e. RR+ )
30	11, 29	syl 16	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> x e. RR+ )
31	21, 30	rpdivcld 11043	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / x ) e. RR+ )
32	31	adantl 466	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / x ) e. RR+ )
33		ppinncl 22511	. . . . . . . . . . 11 |- ( ( x e. RR /\ 2 <_ x ) -> (π ` x ) e. NN )
34	14, 19, 33	syl2anc 661	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. NN )
35	34	nnrpd 11025	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. RR+ )
36	13, 27	rplogcld 22077	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> ( log ` x ) e. RR+ )
37	11, 36	syl 16	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. RR+ )
38	35, 37	rpmulcld 11042	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) e. RR+ )
39	21, 38	rpdivcld 11043	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) e. RR+ )
40	39	adantl 466	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) e. RR+ )
41	30	ssriv 3359	. . . . . . . 8 |- ( 2 [,) +oo ) C_ RR+
42		resmpt 5155	. . . . . . . 8 |- ( ( 2 [,) +oo ) C_ RR+ -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) )
43	41, 42	ax-mp 5	. . . . . . 7 |- ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) )
44		pnt2 22861	. . . . . . . 8 |- ( x e. RR+ |-> ( ( theta ` x ) / x ) ) ~~> r 1
45		rlimres 13035	. . . . . . . 8 |- ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) ~~> r 1 -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
46	44, 45	mp1i 12	. . . . . . 7 |- ( T. -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
47	43, 46	syl5eqbrr 4325	. . . . . 6 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) ~~> r 1 )
48		chtppilim 22723	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ~~> r 1
49	48	a1i 11	. . . . . 6 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ~~> r 1 )
50		ax-1ne0 9350	. . . . . . 7 |- 1 =/= 0
51	50	a1i 11	. . . . . 6 |- ( T. -> 1 =/= 0 )
52	39	rpne0d 11031	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) =/= 0 )
53	52	adantl 466	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) =/= 0 )
54	32, 40, 47, 49, 51, 53	rlimdiv 13122	. . . . 5 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) ~~> r ( 1 / 1 ) )
55	14	recnd 9411	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> x e. CC )
56		chtcl 22446	. . . . . . . . . . . . 13 |- ( x e. RR -> ( theta ` x ) e. RR )
57	13, 56	syl 16	. . . . . . . . . . . 12 |- ( x e. ( 1 (,) +oo ) -> ( theta ` x ) e. RR )
58	57	recnd 9411	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> ( theta ` x ) e. CC )
59	11, 58	syl 16	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) e. CC )
60	55, 59	mulcomd 9406	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( x x. ( theta ` x ) ) = ( ( theta ` x ) x. x ) )
61	60	oveq2d 6106	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) = ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( ( theta ` x ) x. x ) ) )
62	38	rpcnd 11028	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) e. CC )
63	30	rpne0d 11031	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> x =/= 0 )
64	21	rpne0d 11031	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) =/= 0 )
65	62, 55, 59, 63, 64	divcan5d 10132	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( ( theta ` x ) x. x ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
66	61, 65	eqtrd 2474	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
67	38	rpne0d 11031	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) =/= 0 )
68	59, 55, 59, 62, 63, 67, 64	divdivdivd 10153	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) = ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) )
69	34	nncnd 10337	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. CC )
70	37	rpcnd 11028	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. CC )
71	37	rpne0d 11031	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) =/= 0 )
72	69, 55, 70, 63, 71	divdiv2d 10138	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
73	66, 68, 72	3eqtr4d 2484	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) = ( (π ` x ) / ( x / ( log ` x ) ) ) )
74	73	mpteq2ia 4373	. . . . 5 |- ( x e. ( 2 [,) +oo ) |-> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) )
75		1div1e1 10023	. . . . 5 |- ( 1 / 1 ) = 1
76	54, 74, 75	3brtr3g 4322	. . . 4 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 )
77	10, 76	eqbrtrd 4311	. . 3 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
78		ppicl 22468	. . . . . . . . . 10 |- ( x e. RR -> (π ` x ) e. NN0 )
79	13, 78	syl 16	. . . . . . . . 9 |- ( x e. ( 1 (,) +oo ) -> (π ` x ) e. NN0 )
80	79	nn0red 10636	. . . . . . . 8 |- ( x e. ( 1 (,) +oo ) -> (π ` x ) e. RR )
81	29, 36	rpdivcld 11043	. . . . . . . 8 |- ( x e. ( 1 (,) +oo ) -> ( x / ( log ` x ) ) e. RR+ )
82	80, 81	rerpdivcld 11053	. . . . . . 7 |- ( x e. ( 1 (,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. RR )
83	82	recnd 9411	. . . . . 6 |- ( x e. ( 1 (,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. CC )
84	83	adantl 466	. . . . 5 |- ( ( T. /\ x e. ( 1 (,) +oo ) ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. CC )
85		eqid 2442	. . . . 5 |- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) = ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) )
86	84, 85	fmptd 5866	. . . 4 |- ( T. -> ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) : ( 1 (,) +oo ) --> CC )
87	12	a1i 11	. . . 4 |- ( T. -> ( 1 (,) +oo ) C_ RR )
88	15	a1i 11	. . . 4 |- ( T. -> 2 e. RR )
89	86, 87, 88	rlimresb 13042	. . 3 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 <-> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) ~~> r 1 ) )
90	77, 89	mpbird 232	. 2 |- ( T. -> ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 )
91	90	trud 1378	1 |- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1

Syntax hints:   <-> wb 184   /\ w3a 965   = wceq 1369   T. wtru 1370   e. wcel 1756   =/= wne 2605   C_ wss 3327   class class class wbr 4291   e. cmpt 4349   |` cres 4841   ` cfv 5417  (class class class)co 6090   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282   x. cmul 9286   +oocpnf 9414   RR*cxr 9416   < clt 9417   <_ cle 9418   / cdiv 9992   NNcn 10321   2c2 10370   NN0cn0 10578   RR+crp 10990   (,)cioo 11299   [,)cico 11301   ~~> r crli 12962   logclog 22005   thetaccht 22427  πcppi 22430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-disj 4262  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-fi 7660  df-sup 7690  df-oi 7723  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-q 10953  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-ioo 11303  df-ioc 11304  df-ico 11305  df-icc 11306  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-fac 12051  df-bc 12078  df-hash 12103  df-shft 12555  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-limsup 12948  df-clim 12965  df-rlim 12966  df-o1 12967  df-lo1 12968  df-sum 13163  df-ef 13352  df-e 13353  df-sin 13354  df-cos 13355  df-pi 13357  df-dvds 13535  df-gcd 13690  df-prm 13763  df-pc 13903  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-hom 14261  df-cco 14262  df-rest 14360  df-topn 14361  df-0g 14379  df-gsum 14380  df-topgen 14381  df-pt 14382  df-prds 14385  df-xrs 14439  df-qtop 14444  df-imas 14445  df-xps 14447  df-mre 14523  df-mrc 14524  df-acs 14526  df-mnd 15414  df-submnd 15464  df-mulg 15547  df-cntz 15834  df-cmn 16278  df-psmet 17808  df-xmet 17809  df-met 17810  df-bl 17811  df-mopn 17812  df-fbas 17813  df-fg 17814  df-cnfld 17818  df-top 18502  df-bases 18504  df-topon 18505  df-topsp 18506  df-cld 18622  df-ntr 18623  df-cls 18624  df-nei 18701  df-lp 18739  df-perf 18740  df-cn 18830  df-cnp 18831  df-haus 18918  df-cmp 18989  df-tx 19134  df-hmeo 19327  df-fil 19418  df-fm 19510  df-flim 19511  df-flf 19512  df-xms 19894  df-ms 19895  df-tms 19896  df-cncf 20453  df-limc 21340  df-dv 21341  df-log 22007  df-cxp 22008  df-em 22385  df-cht 22433  df-vma 22434  df-chp 22435  df-ppi 22436  df-mu 22437

This theorem is referenced by: (None)